The present invention relates to reducing the errors in the measurement of heat flow rate in a differential scanning calorimeter, i.e., a DSC.
In general a calorimeter may be considered to comprise the measuring system and the enclosure. The measuring system includes the sample, a sample container, if used, and a means to measure the sample temperature Ts. The enclosure encloses the measuring system, isolates it from the environment and regulates the temperature of the calorimeter. In prior art systems, the temperature of the enclosure may be designated by the symbol T0. This temperature is controlled in a manner that depends on the operating principle of the calorimeter and the experimental method. The temperature difference between the enclosure and the measuring system T0−Ts is a measured variable that has been used in different ways depending, mainly, on the operating principle of the calorimeter.
The temperature difference is measured across a thermal resistance between the measuring system and the enclosure. Heat flow within a calorimeter may be described by the equation:
      C    ⁢                  T        .            s        =                              T          0                -                  T          s                    R        +    W  where C is the heat capacity of the sample and its container, if a sample container is used: {dot over (T)}s is the rate of change of temperature of the measuring system with respect to time; R is the thermal resistance between the measuring system and the enclosure; and W is the total of all other heat supplied to or removed from the measuring system. W may include the heat absorbed by or released from the sample during a transition, for example the latent heat of fusion, or it may include heat supplied to or removed from the measuring system as required by the operating mode of the instrument, for example, by heaters that supply power to compensate for sample heat effects.
Calorimeters may be divided into two broad categories depending upon how the temperature difference ΔT=T0−Ts is controlled and used. In adiabatic calorimeters ΔT=0, consequently there is no heat exchange between the measuring system and the enclosure. All other calorimeters where ΔT≠0 may be classified as nonadiabatic because there is heat exchange between the measuring system and the enclosure. Under this classification system, a heat flux differential scanning calorimeter is a nonadiabatic-nonisothermal calorimeter in which the temperature of the enclosure is controlled to follow a desired temperature program, i.e., T0=T0(t). ΔT=ΔT(t), is used as the principal signal in the heat flow rate measurement.
In calorimetry, heat fluxes that occur within the measuring system that are not detected by the heat flow sensor are considered to be heat leakage. Because this heat leakage supplies part of the heat flow between the sample under analysis and the enclosure it may be a measurement error. There are two possibilities for dealing with the problem of heat leakage: adiabatic operation and twin calorimeters.
In adiabatic operation, the temperatures of the measuring system and the calorimeter enclosure are controlled so that they are equal, thereby eliminating heat leakage. In most cases, realization of adiabatic operation requires additional heating or cooling of the measuring system to force ΔT to be zero. Typically electric resistance heating elements and Peltier devices are used in adiabatic calorimeters to heat or cool the measuring system to maintain adiabatic operation.
In twin calorimeters, two nominally identical measuring systems are installed symmetrically within the calorimetric enclosure. One of the calorimeters contains the sample under analysis and the other contains an inert reference sample or is operated empty. To the extent that the two calorimeters are identical and symmetrically placed, the heat leakage of the two will be identical and subtracting the measured heat flow of the reference calorimeter from the sample calorimeter will cancel the heat leakage and heat exchange effects of the measuring systems, such as heat accumulation. However, the presence of the sample means that the two calorimeters are not in fact identical and so, the heat leakage effects and heat exchange effects within the measuring systems are not completely cancelled.
A heat flux differential scanning calorimeter is a twin calorimeter where the measurement of heat flow rate is obtained from the temperature differences between the two measurement systems and the calorimeter enclosure. To get the sample heat flow rate, the principle of conservation of energy is applied to the calorimetric system and an equation or system of equations describing temperature, heat flows and heat inputs is obtained. The resulting equation or set of equations, subject to some level of simplification is used to find the sample heat flow rate from the measured quantities.
A simplified measurement equation for the heat flux DSC may be obtained by assuming steady-state conditions, i.e., constant heat flow rates; only one thermal resistance, the apparent resistance between the furnace and the sample is taken into account assuming no interaction between the sample and reference. Only the heat capacities of the sample and reference (Cs, Cr) are taken into account; the other heat capacities are neglected. The sample temperature and measured temperature are assumed equal and there is no heat exchange with the enclosure, i.e., no heat leakage.
The resulting equation is:
  q  =                    -        Δ            ⁢                          ⁢      T        R  where ΔT=Ts−Tr, Ts and Tr are the temperatures of the sample and reference measuring systems and R is the overall thermal resistance between the sample and the enclosure. This equation is widely used in DSCs presently in use today.
U.S. Pat. No. 6,488,406 (the “'406 patent”), which is incorporated by reference herein, describes a method for measuring heat flow rate in a heat flux DSC that avoids many of the assumptions of the simplified method described above. In particular, it does not assume steady-state conditions. It includes the sample and reference calorimeter thermal resistances and the thermal resistances between the sample and reference calorimeters and their respective containers. It also includes sample and reference container and sample and reference calorimeter heat capacities and sample temperature is not assumed to equal the measured temperature. The measured sample heat flow rate is given by:
  q  =            q      s        -                            m          ps                          m          pr                    ⁢                                    T            .                    ss                                      T            .                    rr                    ⁢              q        r            
The measured sample and reference calorimeter heat flow rates qs and qr are given by:
            q      s        =                            Δ          ⁢                                          ⁢                      T            0                                    R          s                    -                        C          s                ⁢                              T            .                    s                                q      r        =                                        Δ            ⁢                                                  ⁢                          T              0                                +                      Δ            ⁢                                                  ⁢            T                                    R          r                    -                        C          r                ⁡                  (                                                    T                .                            s                        -                          Δ              ⁢                                                          ⁢                              T                .                                              )                                Δ      ⁢                          ⁢              T        0              =                  T        0            -              T        s            where Rs, Rr, Cs and Cr are thermal resistances and heat capacities of the sample and reference calorimeters which are determined by a calibration procedure; mps and mpr are the masses of the sample and reference containers; and {dot over (T)}ss and {dot over (T)}rr are the sample and reference container heating rates.
Sample and reference container temperatures Tss and Trr are given by:Tss=Ts−qsRss Trr=Tr−qrRrr where Rss and Rrr are the thermal contact resistances between the sample and reference containers and their respective calorimeters. Heat flow sensors disclosed in U.S. Pat. No. 6,431,747 (the “'747 patent”) and U.S. Pat. No. 7,470,057 (the “'057 patent”), which are incorporated by reference herein, are suitable for use with this method. These patents disclose means for measuring the two differential temperatures, ΔT and ΔT0, required by the method.
U.S. Pat. No. 7,306,365 (the “'365 patent”), U.S. Pat. No. 7,025,497 (the “'497 patent”) and U.S. Pat. No. 6,843,595 (the “'595 patent”), which are incorporated by reference herein, disclose heat flux differential scanning calorimeters and heat flow rate measurement methods that include heat leakage in the heat flow rate measurement method. In these disclosures it is assumed that the temperature of the DSC enclosure is uniform in temperature and equal to T0, the temperature at the base of the DSC sensor. The equation for sample heat flow including leakage heat flows is:
  q  =                    q        s            ⁡              (                  1          +                                    R              ss                                      R              se                                      )              +                  Δ        ⁢                                  ⁢                  T          0                            R        se              -                            m          ps                          m          pr                    ⁢                                    T            .                    ss                                      T            .                    rr                    ⁢              (                                            q              r                        ⁡                          (                              1                +                                                      R                    rr                                                        R                    re                                                              )                                +                                                    Δ                ⁢                                                                  ⁢                                  T                  0                                            +                              Δ                ⁢                                                                  ⁢                T                                                    R              re                                      )            where Rse, Rss are the thermal resistances between the sample container and the enclosure and between the reference container and the enclosure, i.e., the leakage resistances. This equation is similar in form to the heat flow rate equation of the '406 patent except that it includes two additional terms and two factors multiplying the measured heat flow rates. The second and fourth terms are components of the leakage heat flows between the sample container and the enclosure and between the reference container and the enclosure. The additional factors that multiply the measured sample and reference heat flow rates are each very close to unity because Rse is about two orders of magnitude greater than Rss and Rre is about two orders of magnitude greater that Rrr. The measured sample and reference heat flow rates qs and qr are the same as in the '406 patent.